Wahlquist fluid

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In

Introduction

The Wahlquist fluid was first discovered by Hugo D. Wahlquist in 1968.^[1] It is one of few known exact rotating perfect fluid solutions in general relativity. The solution reduces to the static Whittaker metric in the limit of zero rotation.

Metric

The metric of a Wahlquist fluid is given by

${\displaystyle ds^{2}=f(dt-{\tilde {A}}d\varphi )^{2}-r_{0}^{2}(\zeta ^{2}+\xi ^{2})[{\frac {d\zeta ^{2}}{(1-{\tilde {k}}^{2}\zeta ^{2}){\tilde {h}}_{1}}}+{\frac {d\xi ^{2}}{(1+{\tilde {k}}^{2}\xi ^{2}){\tilde {h}}_{2}}}+{\frac {{\tilde {h}}_{1}{\tilde {h}}_{2}}{{\tilde {h}}_{1}-{\tilde {h}}_{2}}}d\varphi ^{2}]}$

where

${\displaystyle f={\frac {{\tilde {h}}_{1}-{\tilde {h}}_{2}}{\zeta ^{2}+\xi ^{2}}}}$

${\displaystyle {\tilde {A}}=r_{0}({\frac {\xi ^{2}{\tilde {h}}_{1}+\zeta ^{2}{\tilde {h}}_{2}}{{\tilde {h}}_{1}-{\tilde {h}}_{2}}}-\xi _{A}^{2})}$

${\displaystyle {\tilde {h}}_{1}(\zeta )=1+\zeta ^{2}+{\frac {\zeta }{\kappa ^{2}}}[\zeta _{+}{\frac {1}{\tilde {k}}}{\sqrt {1-{\tilde {k}}^{2}\zeta ^{2}}}\arcsin({\tilde {k}}\zeta )]}$

${\displaystyle {\tilde {h}}_{2}(\xi )=1-\xi ^{2}-{\frac {\xi }{\kappa ^{2}}}[\xi _{-}{\frac {1}{\tilde {k}}}{\sqrt {1+{\tilde {k}}^{2}\xi ^{2}}}\sinh ^{-1}({\tilde {k}}\xi )]}$

and ${\displaystyle \xi _{A}}$ is defined by ${\displaystyle {\tilde {h}}_{2}(\xi _{A})=0}$. It is a solution with equation of state ${\displaystyle \mu +3p=\mu _{0}}$ where ${\displaystyle \mu _{0}}$ is a constant.

Properties

The pressure and density of the Wahlquist fluid are given by

${\displaystyle p={\frac {1}{2}}\mu _{0}(1-\kappa ^{2}f)}$

${\displaystyle \mu ={\frac {1}{2}}\mu _{0}(3\kappa ^{2}f-1)}$

The vanishing pressure surface of the fluid is